The dishwasher itself is pretty much all plastic for plumbing and doesn't have any problems with any detergent; it's the dishes that might not react well to specific detergents/temperatures/etc.
The two don't really design for/against each other outside of small stuff like the detergent holder being able to take a detergent pod instead of loose powder. Dishwasher manufacturers are more concerned about form factor and efficient power use while detergent companies are focused on cleaning efficiency.
In my experience, plastics are not immune to damage from “detergent.” Parts of my dishwasher became quite brittle and ultimately failed—and the zip ties used to cobble it back together also become brittle over time.
The two don't really design for/against each other outside of small stuff like the detergent holder being able to take a detergent pod instead of loose powder. Dishwasher manufacturers are more concerned about form factor and efficient power use while detergent companies are focused on cleaning efficiency.